
where \(u \in \mathbb {R}^{p_n}\)  is an additional parameter vector to be optimized and \(\tilde{\beta }\)  is the initial estimator. Note that (REF ) is a linear problem with \(4p_n\)  linear inequality constraints. To prove the existence of ALDS estimates, we need to derive dual problem of (REF ) and prove that strong duality holds. To derive the dual problem, we first construct the Lagrangian form associated with the problem (REF ) considering the main arguments by [1]}
\(L(\beta ; u ; \alpha ) & = \sum _j u_j+ \alpha _1^\top ({\Lambda }_{n} \beta -\mathbf {u})+ \alpha _2^\top (-{\Lambda }_{n} \beta -\mathbf {u})\\& \quad + \alpha _3^\top \Big [ {\mu _n^{-1}} {\Lambda }_{n}^{-1}\Delta _n(\beta )- 1_{p_n} \Big ] + \alpha _4^\top \Big [ - {\mu _n^{-1}} {\Lambda }_{n}^{-1} \Delta _n(\beta ) - 1_{p_n} \Big ]\\& = (1_{p_n} - \alpha _1 - \alpha _2)^\top u + \Big \lbrace  (\alpha _1-\alpha _2)^\top {\Lambda }_{n}- {\mu _n^{-1}} (\alpha _3-\alpha _4)^\top {\Lambda }_{n}^{-1} \mathbf {A}_n( \tilde{\beta })\Big \rbrace \beta \\& \quad + (\alpha _3-\alpha _4)^\top \Big \lbrace  {\mu _n^{-1}} {\Lambda }_{n}^{-1} \Big (\mathbf {U}_{n}( \tilde{\beta }) + \mathbf {A}_n( \tilde{\beta }) \tilde{\beta } \Big ) \Big \rbrace  - (\alpha _3+\alpha _4)^\top 1_{p_n},\) 
